Which equation represents the heat capacity at constant volume (Cv) for a diatomic gas?

Which equation represents the heat capacity at constant volume (Cv) for a diatomic gas?

• A. Cv = (5/2)R
• B. Cv = (3/2)R
• C. Cv = (7/2)R
• D. Cv = (9/2)R

Answer: (A)

The heat capacity at constant volume \(C_v\) for a diatomic gas is derived from the equipartition theorem, which states that energy is equally distributed among all degrees of freedom in a system. A diatomic molecule possesses translational and rotational degrees of freedom. For a diatomic gas, there are: 1. 3 translational degrees of freedom (movement in x, y, and z directions). 2. 2 rotational degrees of freedom (rotation about two axes; the third axis is not counted because it does not contribute significantly to energy at typical temperatures). At room temperature, diatomic gases can also exhibit vibrational modes, but these generally require higher energy to become significantly populated and are often ignored for simplicity in thermodynamic calculations at lower temperatures. Using the equipartition theorem, each degree of freedom contributes \( \frac{1}{2}kT \) (where \(k\) is the Boltzmann constant and \(T\) is the temperature) to the internal energy. For a diatomic gas at room temperature, this results in: - Translational contributions: \(3 \times \frac{1}{2}kT = \frac{3}{2}kT\) - Rotational contributions

The heat capacity at constant volume (C_v) for a diatomic gas is derived from the equipartition theorem, which states that energy is equally distributed among all degrees of freedom in a system. A diatomic molecule possesses translational and rotational degrees of freedom.

For a diatomic gas, there are:

1. 3 translational degrees of freedom (movement in x, y, and z directions).

2. 2 rotational degrees of freedom (rotation about two axes; the third axis is not counted because it does not contribute significantly to energy at typical temperatures).

At room temperature, diatomic gases can also exhibit vibrational modes, but these generally require higher energy to become significantly populated and are often ignored for simplicity in thermodynamic calculations at lower temperatures.

Using the equipartition theorem, each degree of freedom contributes ( \frac{1}{2}kT ) (where (k) is the Boltzmann constant and (T) is the temperature) to the internal energy. For a diatomic gas at room temperature, this results in:

• Translational contributions: (3 \times \frac{1}{2}kT = \frac{3}{2}kT)

• Rotational contributions